Volume 54, Number 2, March-April 2020
|Page(s)||493 - 529|
|Published online||18 February 2020|
On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates
Laboratoire Poems, CNRS/ENSTA/INRIA, Ensta Paris, Université Paris-Saclay, 828, Boulevard des Maréchaux, 91762 Palaiseau, France
2 INRIA/Centre de mathématiques appliquées, École Polytechnique, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau, France
* Corresponding author: firstname.lastname@example.org
Accepted: 15 October 2019
We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.
Mathematics Subject Classification: 35J05 / 35R25 / 35R30 / 65M60
Key words: Cauchy problem / quasi-reversibility / regularity / finite element methods / corners
© The authors. Published by EDP Sciences, SMAI 2020
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